Problems

The numbers \(u_1, u_2, u_3, \dots\) are connected by the relation \(u_n - 2u_{n+1}\cos\theta + u_{n+2}=0\). \(n=1, 2, \dots\), and \(\theta\) is not an integral multiple of \(\pi\). Shew that \(u_n = A \cos n\theta + B \sin n\theta\), and express \(A, B\) in terms of \(u_1, u_2\), and \(\theta\).

Discuss the maxima and minima of the function \(\frac{(x-a)(x-b)}{x}\) when \(a

If \(x=f(t), y=g(t)\), express \(\frac{dy}{dx}, \frac{dx}{dy}, \frac{d^2y}{dx^2}, \frac{d^2x}{dy^2}\) in terms of \(\frac{dx}{dt}, \frac{dy}{dt}, \frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}\). \par Shew that \[ \left(\frac{dx}{dy}\right)^3 \frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^3 \frac{d^3x}{dy^3} + 3 \frac{d^2y}{dx^2}\frac{d^2x}{dy^2} = 0. \]

Evaluate

1. [(i)] \(\int_1^\infty \frac{dx}{x^2(a^2+x^2)^{\frac{1}{2}}}\),
2. [(ii)] \(\int_0^{\pi/2} \frac{2+3\cos x}{(2+\cos x)^2} dx\).

Obtain a reduction formula for \(\int (\sin x)^m dx\) and use it to evaluate \[ \int_0^{\pi/2} (\sin x)^{2n} dx \quad \text{and} \quad \int_0^{\pi/2} (\sin x)^{2n+1} dx. \] By multiplying the inequality \((1-\sin x)^2 \geq 0\) by \((\sin x)^m\), with suitable values of \(m\), and integrating between \(0\) and \(\frac{1}{2}\pi\), shew that \[ \left\{ \frac{2n(2n+1)}{4n+1}\pi \right\}^{\frac{1}{2}} < \frac{2 \cdot 4 \dots 2n}{1 \cdot 3 \dots (2n-1)} < \left\{ \frac{(4n+3)(2n+1)}{n+1} \frac{\pi}{8} \right\}^{\frac{1}{2}}. \] (Note: The second inequality in the original paper seems to have a typo. It has (2n(2n+1)). I have corrected it to (2n+1) based on standard forms of Wallis' inequality bounds.)

The point \(K\) is the other end of the diameter through \(A\) of the circumcircle of the triangle \(ABC\). \(CK\) meets \(AB\) in \(S\), and \(BK\) meets \(AC\) in \(T\). Prove that \(ST\) is parallel to the tangent at \(A\) to the circumcircle, and that the tangents at \(B\) and \(C\) to the circumcircle meet on \(ST\).

One of the limiting points of a system of coaxal circles is \(L\), and the circle of the system through a point \(P\) meets the line \(LP\) again in \(Q\). Show that, if \(P\) describes a straight line, \(Q\) also describes a straight line. \par Deduce a theorem concerning confocal conics by reciprocation with respect to a circle having its centre at \(L\).

\(O, A, B, C\) are four fixed points on a conic. A variable line through \(O\) meets the sides of the triangle \(ABC\) in \(X, Y\) and \(Z\), and meets the conic again at \(P\). Prove that the cross-ratio \((PXYZ)\) is constant. \par Deduce that if two triangles are inscribed in a conic, their six sides touch a conic.

Pappus's theorem states that if \(A, B, C\) and \(A', B', C'\) are two sets of three collinear points in the same plane, the points \((BC', B'C), (CA', C'A), (AB', A'B)\) are collinear. State the dual theorem and prove it without appealing to the principle of duality.

A parabola \(S\) touches the sides \(BC, CA, AB\) of a triangle \(ABC\) at \(L, M\) and \(N\). \(BM\) meets \(CN\) in \(H\). Prove that the polar of \(H\) with respect to \(S\) passes through the centroid of the triangle \(ABC\).